Informally, we can draw an \(\epsilon\)-ribbon about the limit \(f\) and \(f_n\) eventually stays in the ribbon. The same \(\mathcal{N}\) works for every \(x \in E\).

This is the usual convergence in metric space \(\mathcal{C}_b(E)\), the continuous bounded functions on \(E\), where \(d(f, g) = \|f - g\|\).

Theorem.\(\mathcal{C}_b(E)\) is complete when the image is in \(\mathbb{R}^n\).

So we have the Cauchy criterion.

Theorem.\(f_n \xrightarrow{\text{uniform}} f\) on \(E\) if and only if for every \(\epsilon > 0\), there exists \(\mathcal{N}\) such that for all \(m, n > \mathcal{N}\) and \(x \in E\), \(|f_n(x) - f_m(x)| < \epsilon\).

Theorem. If \(f_n \xrightarrow{\text{uniform}} f\) on \(E\) and \(f_n\) is continuous then \(f\) is continuous.